SMS scnews item created by Daniel Hauer at Thu 16 May 2019 1411
Type: Seminar
Distribution: World
Expiry: 20 May 2019
Calendar1: 20 May 2019 1400-1500
CalLoc1: AGR Carslaw 829
CalTitle1: Hardy spaces and Schroedinger operators
Auth: dhauer@p635m2.pc (assumed)

# Hardy spaces and Schroedinger operators

### Marcin Preisner

Macquarie University, Australia
Mon 20th May 2019, 2-3pm, Carslaw Room 829 (AGR)

## Abstract

On ${R}^{d}$ we consider the Schrödinger operator

$Lf\left(x\right)=-\Delta f\left(x\right)+V\left(x\right)f\left(x\right),$

where $\Delta ={\partial }_{{x}_{1}}^{2}+\cdots +{\partial }_{{x}_{d}}^{2}$ and $V\left(x\right)\ge 0$ is a positive function (“potential”).

Let ${T}_{t}=exp\left(-tL\right)$ be the heat semigroup associated with to $L$. In the talk we shall consider the Hardy space

${H}^{1}\left(L\right):=\left\{f\in {L}^{1}\left({ℝ}^{d}\right):\underset{t>0}{sup}{T}_{t}f\left(x\right)\in {L}^{1}\left({ℝ}^{d}\right)\right\}$

which is a natural substitute of ${L}^{1}\left({ℝ}^{d}\right)$ in harmonic analysis associated with $L$. Our main interest will be in showing that elements ${H}^{1}\left(L\right)$ have decompositions of the type $f\left(x\right)={\sum }_{k}{\lambda }_{k}{a}_{k}\left(x\right)$, where ${\sum }_{k}|{\lambda }_{k}|<\infty$ and ${a}_{k}$ (“atoms”) have some nice properties.

In the classical case $V\equiv 0$ on ${ℝ}^{d}$ an atom is a function $a$ for which there exist a ball $B\subseteq {ℝ}^{d}$ such that

$supp\left(a\right)\subseteq B,\phantom{\rule{2em}{0ex}}\parallel a{\parallel }_{\infty }\le |B{|}^{-1},\phantom{\rule{2em}{0ex}}\int a\left(x\right)dx=0.$

We shall see that for $L=-\Delta +V$ we can still prove some atomic decompositions, but the properties of atoms depend on the dimension d and the potential $V$.

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